These results suggest that even if we consider an arbitrarily small value of intensity of selection (i.e. weak selection limit), we may still see fixation probabilities increase with population size for large enough populations \(N>N_{min}(w)\)  (i.e. large population limit). This should not contradict the results obtained in [1]}, since in the scenario we are describing the large population limit should dominate over the weak selection one. Additionally, because turning points \(N_{min}(w)\)  become very large under weak selection, fixation probabilities could become less representative of what happens under anti-coordination games in finite populations. Even though pure states are absorbing ones, when mutations are considered, the population might spend most of the evolutionary time in particular transient states [2]}, [3]}, which can also be called quasi-stationary states [4]}, [5]}, [6]}, [7]}. In appendix , we show that average conditional fixation times increase polynomially with \(N_{min}(w)\)  as they do with \(N\)  under neutral fixation, instead of increasing exponentially as is predicted for anti-coordination games under a fixed intensity of selection [2]}. These results highlight why it is essential to beware the impact of the order in which the weak selection limit \(w\rightarrow 0\)  and the large population limit \(N\rightarrow \infty \)  are considered under evolutionary dynamics, as has been clearly stated in [1]}.
